Questions tagged [algebraic-independence]
28 questions
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Over any field, if some polynomials are algebraically dependent, are their derivatives linearly dependent?
Denote by $P_n$ the space of all polynomials in $n$ variables, with coefficients in a field $\mathbb F$.
A collection of polynomials $(f_1,\cdots,f_m)=\vec f\in P_n\!^m$ is called algebraically dependent if there is some polynomial $g\neq0$ in $P_m$…
mr_e_man
- 5,986
5
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Is $2^{\sqrt2}\cdot\pi$ known to be transcendental number?
Specific Question
I know that $2^{\sqrt2}$ and $\pi$ are each transcendental. But is it known that their product or sum are also transcendental?
Exposition
Schanuel's conjecture is a strengthening of the Lindemann-Weierstrass-theorem. It states
If…
Md Asfaque
- 131
5
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3 answers
Are $\pi$ and $\tan^{-1}\left(2\right)$ rational multiples of each other?
For a proof of quantum universality, I need to show that $\tan^{-1}\left(2\right)$ is not a rational multiple of $\pi$. How do I show this? I feel like showing algebraic independence over the rationals is hard in general, but is it possible for…
416E64726577
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4
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Algebraic Independence of symmetric power sum polynomials
Let $P_k(X_1,...,X_n) = X_1^k+...+X_n^k$. My Question is how to proof that the Polynomials $(P_1,...,P_n)$ are algebraically independent. My first try was to imitate the proof of the algebraic independence in of elementary symmetric functions given…
user1072285
- 681
3
votes
1 answer
Proof check: Existence of algebraically independent real numbers (over the rationals).
I am not too familiar with infinites and I don't trust myself with them, I am not sure if any step I took supposed something that I shouldn't have. Below I am presenting an exercise in the book "Mathematics++" and my proof.
The statement
We recall…
Yuumita
- 569
3
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Linear independence vs probabilistic independence
I am trying to solve the following exercise yet I do not know from which angle to attack it. Therefore, I need a hint or two to get me started on both "if" and "only if" implications.
Let $V$ be a finite-dimensional vector space over a finite field…
qarabala
- 1,131
3
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1 answer
A question on algebraic independence
I just learned about the concept of an algebraic independent set. Here is the definition I started working with.
Let $E/K$ be a field extension. A subset $M \subset E$ is said to be algebraically independent over $K$ if every element $\alpha \in M$…
Richard Kruel
- 407
2
votes
1 answer
Preservation of linear independence under field extension
Let $L/K$ be a field extension and $x \in L$ be transcendental over $K$. Let $A, B$ be $K$-matrices of same size. I want to show that if $A$ has linearly independent columns over $K$, then $Ax+B$ has linearly independent columns over $L$. Here's my…
Bubaya
- 2,408
2
votes
1 answer
$(a,b)$ is algebraically independent over $K$ iff $a$ is algebraically independent over $K$ and $b$ is algebraically independent over $K(a)$
I’m using the following definition of algebraic independence:
If $K \subset L$ is a field extension, and $(\alpha_i)_{i\in I}$ is a subset of elements of $L$, we say that the subset $(\alpha_i)_{i\in I}$ is algebraically independent if the…
dahemar
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Bilinear map and linearly independent set
let $f$ : ${M}\times{M}$ $\mapsto$ $F$ be bilinear map
if $T$ = {$u_{1}$ , $u_{2}$ , .... , $u_{n}$} is a subset of ${M}$
with $f(u_{i} , u_{j}$) = $0$ if i $\neq$ j and otherwise $f(u_{i} , u_{j}$) $\neq$ $0$
show that $T$ is linearly independent…
Liyla morad
- 63
2
votes
0 answers
Vector space spanned by incidence vectors of maximal cliques in a graph
I have a undirected graph $(V,E)$, $V=\{1, \dots, d\}$, with set of maximal cliques $\mathcal C$. I am interested in the subspace of $\mathbb R^d$ spanned by the clique incidence vectors $x_C \in \{0,1\}^d$, $C \in \mathcal C$ where a 1 in place $j$…
Moritz Schauer
- 153
1
vote
2 answers
Suppose $\alpha$ is a root of $8x^{4}+4x+3$, then find the integers $n\in \mathbb{Z}$ such that $n \alpha$ is an algebraic integer.
Suppose $\alpha$ is a root of $8x^{4}+4x+3$, then find the integers $n\in \mathbb{Z}$ such that $n \alpha$ is an algebraic integer.
(1) An algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a…
Rotman1729
- 85
1
vote
2 answers
Do Egyptian representations follow a matroid structure?
Let $V=(v_1,\cdots,v_n), v_i \in \mathbb N$ be independent with respect to $\frac p q$ if there doesn't exist an indicator vector $\beta = (b_1, \cdots, b_n), b_i \in \{0,1\}$ with $\sum \frac{b_i}{v_i} = \frac p q$.
Clearly if $V \subset W$ and…
Snared
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Which general methods are there for determining if $t$ and $e^t$, where $t$ is any given transcendental number, are algebraically independent?
Which general methods are there for determining if $t$ and $e^t$, where $t$ is any given transcendental number, are algebraically independent?
Can Schanuel's conjecture be used for this?
We have Lindemann-Weierstrass theorem, Gelfond-Schneider…
IV_
- 7,902
1
vote
1 answer
Wronskian of x and |x|
I am asked to find the Wronskian of $x$ and $|x|$ in $[-1,1]$.
But $|x|$ is not differentiable at $x=0$.
How do I calculate the Wronskian of such non-differentiable functions?
Could I do this: $W(x,|x|)|_{x\in[-1,0)}+W(x,|x|)|_{x\in(0,-1]}$ because…
Manjoy Das
- 1,066